Application And Study On High Order Scheme Of Gas Kinetic For Solving The Boltzmann Model Equation

Boltzmann equation, as the basic equation of gas kinetic theory, has been proposed in《Further Study on Heat Equilibrium between Gas Molecules》by L. Boltzmann in 1872, and the Boltzmann-H theorem was also proved. But the Boltzmann equation is an integral and differential equation, and the collision term is of nonlinearity and is related to the intermolecular force law. It is very difficult to solve Boltzmann equation directly. Thus, it has been suggested that the Boltzmann equation should be solved by using the simple model equations approximately instead of the original collision term. In studying rarefied gas dynamics with Boltzmann model equation, one of the much successful methods is the gas-kinetic unified algorithm for flows from rarefied transition to continuum which is combined with the discrete velocity ordinate method and CFD finite-difference method. Based on this method, this thesis has introduced and developed several gas-kinetic finite-difference schemes with different precision and some different collision model equations. The effect of these different schemes and models on the results is studied by solving the one-dimension shock wave tube problems and the two-dimension flows around circular cylinder and aerofoil. And the one-dimension shock wave problems in the mixture gases are studied preliminarily using the gas-kinetic unified algorithm.This thesis contains six chapters. Chapter 1 is the preface, in which the research methods of rarefied gas dynamics are introduced simply. Then, the development and actuality of solving Boltzmann model equation is reviewed. At the same time, the research survey of high-order schemes is synopsized. At the end of this chapter, the studying works of this thesis are presented.In chapter 2, the basic principles of gas kinetic theory are introduced. In the following part, the classic constituted process of Boltzmann equation and Boltzmann-H theory are given. The model equations such as BGK, Shakhov and Ellipsoidal Statistical model, are introduced, respectively. The reduced velocity distribution functions are used to cut down the number of independent variables of the Boltzmann model equation. Next, the principle of the discrete velocity ordinate method is presented. Finally, the constructing processes of several high-order compact difference schemes are intrduced.In the third and fourth chapter, the gas-kinetic unified algorithm for one-dimension and two-dimension gas dynamical problems are developed respectively. The effects of these different schemes and models on the results were studied. And the computing efficiency of different schemes and kinetic models is analyzed.In chapter 5, the unified algorithm for one-dimension mixture gas is studied to solve the inner flows of normal shock wave problem; the feasibility and reliablity of the gas-kinetic numerical schemes are investigated to apply in mixture gas flows.The concluding remarks of the thesis are given in Chapter 6. In this chapter, the shortages and unsolved problems of this thesis are also pointed out, and the researching orientations are suggested in the future.